## Abstract

In this paper we consider the problem of best uniform approximation of a real valued semi-Lipschitz function \(F\) defined on an asymmetric metric space \((X,d)\), by the elements of the set \(E_{d}(F|_{Y})\) of all extensions of \(F|_{Y}(Y\subset X)\), preserving the smallest semi-Lipschitz constant. It is proved that, this problem has always at least a solution, if \((X,d)\) is \((d,\overline{d})\)-sequentially compact, or of finite diameter.

## Authors

**Costică Mustăţa**

“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academi, Romania

## Keywords

?

## Paper coordinates

C. Mustăţa, *Best uniform approximation of semi-Lipschitz function by extension*, Rev. Anal. Numér. Théor. Approx. 36 (2007) 2, pp. 161-171.

## About this paper

##### Journal

Revue d’Analyse Numer. Theor. Approx.

##### Publisher Name

Publishing House of the Romanian Academy

##### Print ISSN

2501-059X

##### Online ISSN

2457-6794

google scholar link

[1] Cobzas, S., Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math., 27 (3), pp. 275–296, 2004.

[2] Cobzas, S., Asymmetric locally convex spaces, Int. J. Math. Math. Sci., 16, pp. 2585–2608, 2005.

[3] Cobzas, S. and Mustata, C., Best approximation in spaces with asymmetric norm,. Rev. Anal. Numer. Theor. Approx., 33 (1), pp. 17–31, 2006.

[4] Cobzas, S. and Mustata, C., Extension of bounded linear functionals and best approximation in spaces with asymmetric norm, Rev. Anal. Numer. Theor. Approx., 31, 1, pp. 35–50, 2004.

[5] Collins, J. and Zimmer, J., An asymmetric Arzela-Ascoli theorem, http://bath.ac.uk/math-sci/BICS, Preprint, 16, 12 pp, 2005.

[6] Garcia-Raffi, L. M., Romaguera, S. and Sanchez-Perez, E. A., The dual space of an asymmetric linear space, Quaest. Math., 26, pp. 83–96, 2003.

[7] Kunzi, H. P. A., Nonsymmetric distances and their associated topologies: about the origin of basic ideas in the area of asymmetric topologies, in: Handbook of the History of General Topology, ed. by C.E. Aull and R. Lower, 3, Hist. Topol. 3, Kluwer Acad. Publ. Dordrecht, pp. 853–968, 2001.

[8] Mc.Shane, E. T., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837–842, 1934.

[9] Menucci, A., On asymmetric distances, Technical Report, Scuola Normale Superiore, Pisa, 2004.

[10] Mustata, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx, 30, 1, pp. 61–67, 2001.

[11] Mustata, C., On the extremal semi-Lipschitz functions, Rev. Anal. Numer. Theor. Approx, 31, 1, pp. 103–108, 2002.

[12] Mustata, C., On the approximation of the global extremum of a semi-Lipschitz function, IJMMS (to appear).

[13] Reilly, I.L., Subrahmanyam, P. V. and Vamanamurthy, M. K., Cauchy sequences in quasi-pseudo-metric spaces, Mh. Math., 93 , pp. 127–140, 1982.

[14] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292–301, 2000.

[15] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitz functions, Acta Math. Hungar., 108(1-2), pp. 55–70, 2005.

[16] Romaguera, S., Sanchez-Alvarez, J.M. and Sanchis, M., El espacio de funciones semi-Lipschitz, VI Jornadas de Matematica Aplicada, Universiadad Politecnica de Valencia, pp. 1–15, 2005.

[17] Sanchez-Alvarez, J. M., On semi-Lipschitz functions with values in a quasi-normed linear space, Applied General Topology, 6, 2, pp. 216–228, 2005.